Environmental Engineering Reference
In-Depth Information
Nevertheless, the reliability is significantly higher following the measurements. The reason
is the reduced uncertainty, which can be observed in
Figure 5.2.
The probability of values
μ
φ
< 21.36° is reduced from the prior to the posterior distribution.
if
X
;
more precisely, they are expected values. This includes the mean value and covariance of
the uncertain parameters
X
as well as the probability of failure. It follows that prediction
requires methods that can perform such integrations efficiently, similar to the computation
of the constant
a
. This will be discussed in Section 5.3.5.
As in the example above, the quantities of interest are generally integral functions of
′′
5.3 geoteChnICal relIabIlItY baSeD on MeaSureMentS:
SteP-bY-SteP ProCeDure For baYeSIan analYSIS
5.3.1 Initial probabilistic model: Prior distribution
The first step in the analysis is to establish the geotechnical model, that is, the ultimate and
serviceability LSFs, and to propose an initial probabilistic model of its parameters. This
is the prior distribution
′
if
X
.
Once the relevant uncertain parameters
X
are identified, their
probability distributions are established as in classical geotechnical reliability analysis (e.g.,
Rackwitz 2000). Thereby, all relevant information prior to making the measurements is col-
lected and assessed and appropriate models are found.
In many instances, information on the parameters is available from knowledge on the
geology or geotechnical conditions at nearby sites. Information may also be available from
the previous geotechnical assessments, as well as from earlier measurements at the site.
Additionally, literature sources can provide probabilistic models of soil conditions. Finally,
expert estimations can and should also be included, but thereby, it has to be ensured that
information is not used twice. (The expert's estimate may be founded in past measurements.
If these measurements are considered as additional separate independent information, then
the information content is overestimated, resulting in overconfidence.) When combining dif-
ferent sources of information, it is also possible to use Bayesian analysis, whereby one source
of information (e.g., literature) is taken as a prior and the remaining information (e.g., previ-
ous measurements) are modeled by the likelihood function following Section 5.3.3.
If the available information is vague, then the prior distribution should reflect this. In
Bayesian analysis, one often uses the so-called “non-informative priors,” that can, for
example, be a uniform distribution from 0 to ∞ for a non-negative real-valued parameter
Y
. However, the term “non-informative prior” is misleading, since it is not possible to put
no information into a prior distribution. For example, the prior distribution for the non-
negative real-valued parameter
Y
might also be chosen as a uniform distribution on log
Y
,
defined from −∞ to ∞; clearly, this would differ from the choice above, but both choices seem
to reflect ignorance on the value of this parameter. For this reason, a number of different
definitions of noninformative priors have been proposed in the mathematical literature, for
example, using the maximum entropy principle (Jaynes 1968) or the so-called reference prior
(Bernardo 1979). The latter is defined as the prior that maximizes the expected Kullback-
Leibler (KL) divergence of the posterior distribution with respect to the prior distribution.
Since the KL divergence is a measure of information gain, this will lead to the choice of the
prior distribution with the least information relative to the posterior. This definition of the
noninformative prior leads to different prior distributions for different likelihood functions.
We note that for engineering applications, the discussion on what constitutes a non-
informative prior is of little practical relevance. First, one almost always has some, if only
Search WWH ::
Custom Search